L(s) = 1 | + (−0.670 + 0.742i)2-s + (−0.101 − 0.994i)4-s + (0.557 − 0.830i)5-s + (0.970 + 0.242i)7-s + (0.806 + 0.591i)8-s + (0.242 + 0.970i)10-s + (0.998 − 0.0611i)11-s + (−0.794 − 0.607i)13-s + (−0.830 + 0.557i)14-s + (−0.979 + 0.202i)16-s + (0.540 + 0.841i)17-s + (0.994 − 0.101i)19-s + (−0.882 − 0.470i)20-s + (−0.623 + 0.781i)22-s + (−0.377 − 0.925i)25-s + (0.983 − 0.182i)26-s + ⋯ |
L(s) = 1 | + (−0.670 + 0.742i)2-s + (−0.101 − 0.994i)4-s + (0.557 − 0.830i)5-s + (0.970 + 0.242i)7-s + (0.806 + 0.591i)8-s + (0.242 + 0.970i)10-s + (0.998 − 0.0611i)11-s + (−0.794 − 0.607i)13-s + (−0.830 + 0.557i)14-s + (−0.979 + 0.202i)16-s + (0.540 + 0.841i)17-s + (0.994 − 0.101i)19-s + (−0.882 − 0.470i)20-s + (−0.623 + 0.781i)22-s + (−0.377 − 0.925i)25-s + (0.983 − 0.182i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.640163185 + 0.05518616995i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.640163185 + 0.05518616995i\) |
\(L(1)\) |
\(\approx\) |
\(1.045755559 + 0.1225813569i\) |
\(L(1)\) |
\(\approx\) |
\(1.045755559 + 0.1225813569i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.670 + 0.742i)T \) |
| 5 | \( 1 + (0.557 - 0.830i)T \) |
| 7 | \( 1 + (0.970 + 0.242i)T \) |
| 11 | \( 1 + (0.998 - 0.0611i)T \) |
| 13 | \( 1 + (-0.794 - 0.607i)T \) |
| 17 | \( 1 + (0.540 + 0.841i)T \) |
| 19 | \( 1 + (0.994 - 0.101i)T \) |
| 31 | \( 1 + (0.965 - 0.262i)T \) |
| 37 | \( 1 + (-0.359 - 0.933i)T \) |
| 41 | \( 1 + (0.989 + 0.142i)T \) |
| 43 | \( 1 + (-0.965 - 0.262i)T \) |
| 47 | \( 1 + (0.974 + 0.222i)T \) |
| 53 | \( 1 + (-0.0203 + 0.999i)T \) |
| 59 | \( 1 + (-0.415 + 0.909i)T \) |
| 61 | \( 1 + (0.0407 + 0.999i)T \) |
| 67 | \( 1 + (0.0611 - 0.998i)T \) |
| 71 | \( 1 + (0.488 - 0.872i)T \) |
| 73 | \( 1 + (0.122 - 0.992i)T \) |
| 79 | \( 1 + (-0.202 + 0.979i)T \) |
| 83 | \( 1 + (-0.685 + 0.728i)T \) |
| 89 | \( 1 + (0.396 + 0.917i)T \) |
| 97 | \( 1 + (-0.505 - 0.862i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.01792880257443865140681544365, −19.00328090284567291447193803902, −18.63834249037624001703601346731, −17.7078723000809665089032160192, −17.33973785180889921718880889506, −16.65718507613739691831718115899, −15.66373710636955174289219701787, −14.446251284799218463780111172731, −14.18892483224288020373481736695, −13.430482165578130734342057259341, −12.17942920168866824501140158347, −11.638998726531739763485567188709, −11.18316242576921016407933058735, −10.074912917700286499855984854, −9.75217174195664514258424062368, −8.912465516633703790751721535822, −7.93306639479746386878108692350, −7.194152962195480298049942679545, −6.644862433060171060446085450112, −5.295453329264873250284934151571, −4.46518381845489244655391447, −3.48178505782538198564893949509, −2.65994460973051437549819139319, −1.79504322499167191249749974903, −1.04426073275565002539619967266,
0.917773824827270935549958718483, 1.50164883861093452703420718428, 2.49907106583939811733206419619, 4.09178364434060443630223846978, 4.91047811142209089857849802422, 5.58358529457710036123011791586, 6.18054810781446896007006479739, 7.36173321725175025593855473021, 7.95181044572485682047761180180, 8.73538979964682578159095310905, 9.33976440778388486212941881953, 10.058354060703720268164520644304, 10.87138326489463794516341804956, 11.902412673606034073880368160174, 12.46816517699899803600198882228, 13.72919913311414897327345675888, 14.14479815603925883232868196670, 14.97887626978429613392701249837, 15.54566557684885075237573776707, 16.70909342109229067410471819914, 16.9441205734156675996851294483, 17.74226452614680251127446061972, 18.128406709514805708221792208350, 19.28168953038710521730597464404, 19.812963674423999005601646995965